L(s) = 1 | − 2-s + 3-s + 4-s + 1.95·5-s − 6-s − 2.89·7-s − 8-s + 9-s − 1.95·10-s + 3.65·11-s + 12-s + 13-s + 2.89·14-s + 1.95·15-s + 16-s − 5.20·17-s − 18-s + 4.31·19-s + 1.95·20-s − 2.89·21-s − 3.65·22-s − 0.240·23-s − 24-s − 1.17·25-s − 26-s + 27-s − 2.89·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.874·5-s − 0.408·6-s − 1.09·7-s − 0.353·8-s + 0.333·9-s − 0.618·10-s + 1.10·11-s + 0.288·12-s + 0.277·13-s + 0.773·14-s + 0.504·15-s + 0.250·16-s − 1.26·17-s − 0.235·18-s + 0.989·19-s + 0.437·20-s − 0.631·21-s − 0.778·22-s − 0.0501·23-s − 0.204·24-s − 0.235·25-s − 0.196·26-s + 0.192·27-s − 0.547·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.068523079\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.068523079\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 + 0.240T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 - 9.50T + 41T^{2} \) |
| 43 | \( 1 + 9.20T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 + 1.79T + 53T^{2} \) |
| 59 | \( 1 - 0.347T + 59T^{2} \) |
| 61 | \( 1 - 4.16T + 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 5.67T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.987223345785685281905300840092, −7.03552402671925560361465397794, −6.47083990179404004594281280332, −6.17855149299642165344383562297, −5.11612347697229752132385069102, −4.06835502469044754485759907712, −3.34215929174341611234221646277, −2.54215840136098383732789613903, −1.77744818463544824575736054740, −0.77907956562831183226219071087,
0.77907956562831183226219071087, 1.77744818463544824575736054740, 2.54215840136098383732789613903, 3.34215929174341611234221646277, 4.06835502469044754485759907712, 5.11612347697229752132385069102, 6.17855149299642165344383562297, 6.47083990179404004594281280332, 7.03552402671925560361465397794, 7.987223345785685281905300840092